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Proteins analysed as virtual knots

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ABSTRACT

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, which are a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.

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Classical and virtual knot diagrams.(a) The first six classical knots in the standard tabulation (including the unknot 01); all but 51 have been identified as dominant knot types in at least one protein under sphere closure5. (b) The virtual knots with n = 2,3,4 as tabulated in ref. 20, all of which can arise as virtual closures of open knot diagrams (i.e. the minimally genus one virtual knots, described in Supplementary Note 1). Virtual crossings are shown as circles. (c–h) show examples of open diagrams, which may be identified under virtual closure as classical or virtual knots. (c–e) are equivalent to the projections from Fig. 1(d). (f) and (g) show (e) closed with a classical arc passing above or below the intervening strands, forming an unknot 01 and trefoil knot 31 respectively, while (h) shows (e) closed instead with a virtual crossing to produce the knot v21.
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f2: Classical and virtual knot diagrams.(a) The first six classical knots in the standard tabulation (including the unknot 01); all but 51 have been identified as dominant knot types in at least one protein under sphere closure5. (b) The virtual knots with n = 2,3,4 as tabulated in ref. 20, all of which can arise as virtual closures of open knot diagrams (i.e. the minimally genus one virtual knots, described in Supplementary Note 1). Virtual crossings are shown as circles. (c–h) show examples of open diagrams, which may be identified under virtual closure as classical or virtual knots. (c–e) are equivalent to the projections from Fig. 1(d). (f) and (g) show (e) closed with a classical arc passing above or below the intervening strands, forming an unknot 01 and trefoil knot 31 respectively, while (h) shows (e) closed instead with a virtual crossing to produce the knot v21.

Mentions: We now summarise some basic mathematics of knot and virtual knot classification813. A more complete summary of both classical and virtual knot theory is given in Supplementary Note 1. Knots are labelled and ordered in knot tables14151617 according to their minimal crossing number n, which is the minimum number of crossings a 2-dimensional diagram of the knot may have8. The closed knots with n crossings are labelled nm, where m is an effectively arbitrary index, not distinguishing enantiomeric pairs with opposite chirality (our analysis does not distinguish between such pairs, although it would be possible to do so). Some simple knots are shown in Fig. 2(a) such as the unknot 01 (counted for completeness) and the trefoil knot 31 (the only knot with n = 3). Composite knots, in which more than one knot is tied in a single curve, do not appear in protein chains5. A given knot has many possible conformations, which may have arbitrarily many crossings in projection. Equivalent conformations, which can be deformed into one another without cutting and joining, are called ambient isotopic; their diagrams can be related algorithmically by a sequence of Reidemeister moves, a set of local arc and crossing changes representing smooth deformation of a 3D curve8 (see Supplementary Fig. 1).


Proteins analysed as virtual knots
Classical and virtual knot diagrams.(a) The first six classical knots in the standard tabulation (including the unknot 01); all but 51 have been identified as dominant knot types in at least one protein under sphere closure5. (b) The virtual knots with n = 2,3,4 as tabulated in ref. 20, all of which can arise as virtual closures of open knot diagrams (i.e. the minimally genus one virtual knots, described in Supplementary Note 1). Virtual crossings are shown as circles. (c–h) show examples of open diagrams, which may be identified under virtual closure as classical or virtual knots. (c–e) are equivalent to the projections from Fig. 1(d). (f) and (g) show (e) closed with a classical arc passing above or below the intervening strands, forming an unknot 01 and trefoil knot 31 respectively, while (h) shows (e) closed instead with a virtual crossing to produce the knot v21.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5304221&req=5

f2: Classical and virtual knot diagrams.(a) The first six classical knots in the standard tabulation (including the unknot 01); all but 51 have been identified as dominant knot types in at least one protein under sphere closure5. (b) The virtual knots with n = 2,3,4 as tabulated in ref. 20, all of which can arise as virtual closures of open knot diagrams (i.e. the minimally genus one virtual knots, described in Supplementary Note 1). Virtual crossings are shown as circles. (c–h) show examples of open diagrams, which may be identified under virtual closure as classical or virtual knots. (c–e) are equivalent to the projections from Fig. 1(d). (f) and (g) show (e) closed with a classical arc passing above or below the intervening strands, forming an unknot 01 and trefoil knot 31 respectively, while (h) shows (e) closed instead with a virtual crossing to produce the knot v21.
Mentions: We now summarise some basic mathematics of knot and virtual knot classification813. A more complete summary of both classical and virtual knot theory is given in Supplementary Note 1. Knots are labelled and ordered in knot tables14151617 according to their minimal crossing number n, which is the minimum number of crossings a 2-dimensional diagram of the knot may have8. The closed knots with n crossings are labelled nm, where m is an effectively arbitrary index, not distinguishing enantiomeric pairs with opposite chirality (our analysis does not distinguish between such pairs, although it would be possible to do so). Some simple knots are shown in Fig. 2(a) such as the unknot 01 (counted for completeness) and the trefoil knot 31 (the only knot with n = 3). Composite knots, in which more than one knot is tied in a single curve, do not appear in protein chains5. A given knot has many possible conformations, which may have arbitrarily many crossings in projection. Equivalent conformations, which can be deformed into one another without cutting and joining, are called ambient isotopic; their diagrams can be related algorithmically by a sequence of Reidemeister moves, a set of local arc and crossing changes representing smooth deformation of a 3D curve8 (see Supplementary Fig. 1).

View Article: PubMed Central - PubMed

ABSTRACT

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, which are a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.

No MeSH data available.


Related in: MedlinePlus